Patterns of Flavour Violation in a Warped Extra Dimensional Model with Custodial Protection

0.66 MB·English

by  Stefania Gori

9
Patterns of Flavour Violation in a Warped Extra
0
0
2 Dimensional Model with Custodial Protection

n
a
J Stefania Gori

9
Physik Department, TechnischeUniversit¨at Mu¨nchen, D-85748 Garching, Germany
2
E-mail: [email protected]

]
h
p Abstract. We present a particular warped extra dimensional model, where the flavour
- diagonal andflavournon-diagonalZ boson couplingstoleft-handeddownquarksareprotected
p
e by the custodial symmetry PLR. After a brief introduction of the model and of its main
h theoretical motivations, we present a complete studyof rare K and B meson decays, including
[ K+ → π+νν¯, KL → π0νν¯, Bs,d → µ+µ− and Bs,d → Xs,dνν¯. In particular we restrict

the parameter space of the model to the subspace which fits all quark masses, CKM mixing
1
parameters and all the measured ∆F = 2 observables, keeping the Kaluza-Klein scale in the
v
reach of LHC (∼ (2−3)TeV). There we show that, in addition to the one loop contribution
4
of the Standard Model (SM), the dominating new physics contribution to the rare decays of
0
7 K and Bs,d mesons is the tree level exchange of the Z boson of the SM governed by right-
4 handedcouplingstodown-typequarks. Inordertoreducetheparameterdependence,westudy
. correlations between various branching ratios of B and K mesons and between ∆F = 1 and
1
∆F = 2 observables. The patterns that we find allow to distinguish this new physics scenario
0
from the SM and can offer an opportunity to future experiments to confirm or rule out the
9
0 model.
:
v
i
X
r
a 1. Introduction
Models with a warped extra dimension (WED) [1], in which all the Standard Model (SM) fields
are allowed to propagate in the bulk, offer a natural solution, or at least alleviation, to some
important puzzles of particle physics. As important examples we can mention the problem
of the gauge hierarchy, the problem in generating the measured large hierarchies in masses and
mixingsofthefermionsoftheSMandinsuppressingFlavourChangingNeutralCurrent(FCNC)
interactions. Of first interest is the study of the flavour changing transitions in the context of
these type of models. What we intend to present here are the results obtained in [2] for the rare
decays of K and B mesons taking into account all the constraints already found analysing the
∆F = 2 transitions [3] (in particular the constraint from the well measured ǫ observable, in
K
combination with a not too large fine tuning) and imposing a Kaluza-Klein scale in the reach of
LHC((2−3)TeV). Inparticular wewant toshow that, once wehave fixedǫ in accordance with
K
the experiments, the model predicts some testable patterns and correlations not only between
various ∆F = 1 observables but also between ∆F = 1 and ∆F = 2 observables.
2. Why Warped Extra Dimensions?
There are many reasons for which one could be led to study warped extra dimensional models.
To analyse all these motivations is clearly beyond the scope of this work. We will just present
twoofthemostimportant: theproblemofthegaugehierarchyandhowtogenerate theordinary
fermion mass and mixing hierarchies in the context of WED.
2.1. The Gauge Hierarchy Problem
It is natural to assume that there is a theory beyond the Standard Model which completes the
SM at higher energy scales (Λ ) and which reproduces at energy scales much lower than Λ
NP NP
theparticle content of the SM.In thisBeyond theStandardModel(BSM) theory wewould have
two scales: the electroweak symmetry breaking (EWSB) scale (the vacuum expectation value
of the Higgs boson v) or equivalently the mass of the Higgs boson M and the scale of new
H
physics (Λ )1. This type of assumption leads to the gauge hierarchy problem: how to explain
NP
that we have this enormous separation between the two scales? The problem would be already
grave at tree level, but it is even worse if we consider one loop corrections to the mass of the
Higgs, since we have in general enormous quantum corrections to the mass of the Higgs from
every particle which couples directly or indirectly to the Higgs field, pushing M closer to the
H
high energy scale.
The most popular way to address the problem is supersymmetry. Here we want to show
briefly how we could solve it just adding an extra dimension to the four usual ones, and putting
in this space a non flat metric.
The main idea of WED [1] is to consider a space (the bulk) with five dimensions. In the
bulk, instead of putting the Minkowski metric, we put a solution of the 5 dimensional Einstein
equations
ds2 = e−2kyη dxµdxν −dy2, (1)
µν
where η = diag(1,−1,−1,−1) is the usual Minkowski metric and k a free parameter, fixed
µν
only by the phenomenology.
The additional dimension is compactified on the S1/Z2 orbifold, generating two Lorentz
invariant branes, one at y = 0 (called Planck or UV brane), the other at y = L (called SM or
IR brane). Regarding the field content, in the simplest WED scenario the Higgs field of the SM
is put on the IR brane, while gravity lives on the UV brane. So it is natural to assume that the
typical energy scales on the two branes are the Planck mass (M ) on the UV brane and the
pl
electroweak scale (v) on the IR brane.
The background metric that we have introduced in the bulk is responsible for solving or at
least alleviating the gauge hierarchy problem [1]. Indeed, if we consider a fundamental energy
scale of Λ on the UV brane, then we receive, going toward the IR brane, an exponential
fund
suppression, due to the warping factor e−ky. In particular, if we compute the evolution of the
fundamental energy scale until the IR brane, we find
Λ (L) = e−kLΛ . (2)
eff fund
Since the Higgs boson lives on the IR brane, this effective energy scale can be seen naturally as
the EWSB scale v.
So, thanks to the exponential factor present in the metric, in order to generate a large
hierarchy between Λ and Λ (L), we do not need an extremely large exponent kL. In
fund eff
numbers, if we take Λ equal to M , we just need kL ≈ 30.
fund pl
So the warped metric in the bulk allows us to see the EWSB scale just as the exponential
suppression of the Planck scale with a quite natural value for the warping factor. In this way
we alleviate the problem of the gauge hierarchy.
1 If we want a theory which includes gravity,we haveto takeas high energy scale the Planck scale. In this case
v/ΛNP ≈10−16.
2.2. The Flavour Problem
Warpedextradimensionalmodelscanalsoprovidenaturallyasolutiontothefollowingquestion:
Why are the masses and mixings of fermions of the SM so hierarchical? To answer this question
we need some more basics on WED. In these type of models gauge and matter fields can
propagate into the fifth dimension. Every time that we solve the equations of motion for the
fields, instead of finding a unique solution, we find an infinity of particles: the Kaluza-Klein
tower (KK) [4] of particles. If a tower has got a zero mode, that is a mode with vanishing mass,
this particle is a particle of the SM. All the other more massive particles are KK excitations of
the particles of the SM.
In particular, the bulk profiles of left-handed and right-handed fermionic zero modes depend
strongly on their so called bulk mass parameters c
(1−2c)kL
f(0)(y,c) = e−cky, f(0)(y,c) = f(0)(y,−c), (3)
L se(1−2c)kL−1 R L
withrespecttothewarpedmetric. Weremarkthatingeneralthebulkmassescarenotuniversal
(0)
for different fermion flavours. Using these shape functions f (y,c) and the Higgs SM doublet
L,R
residing on the IR brane, we can write down the effective 4D Yukawa couplings
ekL
Yu,d = λu,d f(0)(L,ci )f(0)(L,cj ), (4)
ij ij kL L Q R u,d
where λu,d are the fundamental 5D Yukawa coupling matrices. Due to the exponential
dependence of Yu,d on the bulk mass parameters c , the strong hierarchies of quark masses
Q,u,d
and mixings can be traced back to O(1) bulk masses and anarchic 5D Yukawa couplings λu,d.
In numbers, if we choose completely anarchic 5D Yukawas, warping factor kL ≈ 30 and bulk
masses of quarks with different flavour differing by 50%, we find 4D Yukawas differing by two
orders of magnitude. In conclusion, the hierarchies of 4D Yukawas, and so of quark masses and
mixings, is explained naturally by a pure geometrical approach, leading to a partial solution of
the flavour problem. What is still missing would be a theory able to explain the several values
for the bulk masses.
3. The Model
3.1. The Field Content
We consider an SU(3) ×SU(2) ×SU(2) ×U(1) ×P gauge theory on a slice of AdS [1]
c L R X LR 5
with the metric given in (1). The fifth coordinate is restricted to the interval 0 ≤ y ≤ L, and
the KK scale lowered to 2.5TeV in the reach of LHC2.
In the electroweak sector, we consider the gauge symmetry [6–8]
O(4)×U(1) ∼ SU(2) ×SU(2) ×P ×U(1) , (5)
X L R LR X
where P is the discrete symmetry interchanging the two SU(2) groups. These two additional
LR
groups of symmetries, SU(2) × P , are put in order to have custodial protection of the T
R LR
parameter [7] and of the Zb ¯b coupling [8]34. The gauge group in (5) is broken to the SM one
L L
on the UV brane (y = 0), i.e.
SU(2) ×SU(2) ×P ×U(1) → SU(2) ×U(1) (6)
L R LR X L Y
2 Seetheoriginal paper[3]andthewriteup [5]forthedetails onhowtocombinethislow highenergyscale with
theconstraint coming from theobservable ǫK.
3 In the following we will see the generalization of this protection to couplings that are not flavourdiagonal.
4 An analysis of the flavour and electroweak sector in a WED model without custodial protection can be found
in [9].
and on the IR brane to:
SU(2) ×SU(2) ×P ×U(1) → SU(2) ×U(1) . (7)
L R LR X V X
From the enlarged gauge group there arise three new neutral electroweak gauge bosons
Z , Z′, A(1) (8)
H
in addition to the SM Z boson and photon, where the first two are linear combinations of the
gauge eigenstates Z(1) and Z(1) [10]. Neglecting small SU(2) breaking effects on the UV brane
X R
and corrections due to electroweak symmetry breaking, one finds
MZH = MZ′ = MA(1) ≡ MKK ≈ 2.5TeV. (9)
Without entering too much into details for the fermion sector, we give the main features
which will be useful in the following when we will study the rare decays of K and B mesons. In
order to preserve the discrete P symmetry, we decide to embed all the three generations of
LR
left-handed SM down quarks into a symmetric representation of the discrete symmetry. In this
way, also the off-diagonal couplings Zdi d¯j are protected [3] by the same mechanism seen in [8]
L L
for the diagonal Zb ¯b coupling.
L L
3.2. FCNC at Tree Level
Due to the non universality of the shape functions of the zero modes of fermions (3), we have a
different localization of the fermions with different flavour in the bulk, depending on their bulk
masses. In particular heavy fermions will reside close to the IR brane, instead light fermions are
closer to the UV brane.
A different behaviour is shown by the gauge bosons. Solving their equation of motion, we
findthat the zero modes have a flat shapefunction, instead the KK modes are localized towards
the IR brane.
Schematically we can write the interaction between SM fermions and a generic gauge boson
as
L 2
∆ ∝ dyeky f(0) (y,ci ) g(y), (10)
L,R L,R Ψ
Z0 h i
where g(y) is the shape function of the gauge boson in question.
As a consequence of the localization of the gauge KK modes towards the IR brane, their
couplings to zero mode fermions are not flavour universal, but depend strongly on the relevant
bulk mass parameters ci . After rotation to the fermion mass eigenbasis, then, in general
Q,u,d
complex, flavour changing couplings of the heavy gauge bosons Z , Z′ and A(1) are induced.
H
Additionally, duetothemixingof theSMZ bosonwith theheavy KKmodesZ(1) andZ(1), also
X
theZ bosoncouplingsbecomeflavourviolatingalreadyattreelevel. Duetothesenewsourcesof
flavour and CPviolation beyond theCKMmatrix, themodelis clearly a modelbeyondMinimal
Flavour Violation (MFV) [11–13].
4. Rare Decays: Theoretical Background
From this section on we study the rare decays of K and B mesons. In particular we focus on
the decays5
5 For the complete analysis that includes also the rare decays B → Kν¯ν, KL → π0l+l−, KL → µ+µ− see the
original paper [2].
K+ → π+ν¯ν; K → π0ν¯ν; B → X ν¯ν; B → µ+µ−. (11)
L s,d s
To clarify in what our analysis consists of, we consider the first decay in (11). As it is well
know, in the SM this process K+ → π+ν¯ν does not arise at tree level, but only at the one-loop
level.
s ν
Z,Z′,Z
H
d ν
Figure 2. Tree level contributions
of Z, Z′ and Z to the s → dνν¯
H
Figure 1. Feynman diagrams responsible for effective hamiltonian.
the decay K+ → π+ν¯ν in the SM (from [14]).
If we consider the Feynman diagrams in Fig. 1 as effective vertices for the decay in question,
we find the following top quark contribution to the effective hamiltonian6
HSM ∝ V∗V X (s¯γ (1−γ )d)(ν¯γµ(1−γ )ν) . (12)
eff ts td SM µ 5 5
At this point some comments are worthwhile:
• we have dependence just on an operator of the type (V −A)⊗(V −A);
• the function X is a real and universal function in flavour space;
SM
• all the quark dependence is factorized in a prefactor that is the product of two elements of
the CKM matrix (V∗V ).
ts td
In WED the pattern for this decay has some new aspects. As already seen in Sec. 3.2, we
haveFCNCtransitionsalready attreelevel mediatedbytheZ bosonoftheSMandbytheother
heavy KK gauge bosons Z′ and Z . So we have the additional Feynman diagrams at tree level
H
shown in Fig. 2 to add to those of the SM. Computing this new class of Feynman diagrams, we
find for the effective hamiltonian two contributions: one is of the same type of that one already
found in the SM (12) (V∗V XV−A(s¯γ (1−γ )d)(ν¯γµ(1−γ )ν)) with a function XV−A that
ts td µ 5 5
is now complex and not universal anymore, and an additional one of the type
Hnew ∝ V∗V XV (s¯γ d)(ν¯γµ(1−γ )ν) . (13)
eff ts td µ 5
Therefore,
• we have a new operator involved of the type V ⊗(V −A);
• the function XV is complex and not universal.
6 For simplicity we omitted the SM contribution of the charm quark, which is however relevant in the process
K+ →π+ν¯ν.
Interesting is also to notice that, computing all the three diagrams in Fig. 2, the main
contribution is due to the coupling of the Z boson of the SM to the right-handed down quarks.
Instead the coupling of Z to left-handed down quarks is strongly suppressed because of the
protection already discussed in Sec. 3.
4.1. Bigger Contribution of New Physics in K or B Decays?
Summing the contribution of the SM and the new contributions of WED, we can write down in
all generality the contribution of the top quark to the effective hamiltonian for the elementary
process q → q ν¯ν
1 2
Htot ∝ V∗ V X +XV−A (q¯γ (1−γ )q )(ν¯γµ(1−γ )ν)+
eff tq1 tq2 SM q1,q2 1 µ 5 2 5
+ Vt∗q1Vtq2X(cid:0)qV1,q2(q¯1γµq2)((cid:1)ν¯γµ(1−γ5)ν) , (14)
where we can express the new functions XV−A and XV as
q1,q2 q1,q2
1 1
XV−A,V ∝ FV−A,V ∆νν,∆q1,q2 ≡ FV−A,V ∆νν,∆q1,q2 . (15)
q1,q2 V∗ V L L,R λq1,q2 L L,R
tq1 tq2 (cid:16) (cid:17) t (cid:16) (cid:17)
Since the functions F depend on the quarks involved in the decay, the flavour of the quarks
does not enter only through the dependence on the elements of the CKM matrix. This should
becontrasted to the case of the SM where these decays are governed by a flavour-universal loop
function X and the only flavour dependence enters through the CKM factors.
SM
Now we specialize this effective hamiltonian to the case of s → dν¯ν and b → (d,s)ν¯ν, for
the decays of K and B mesons respectively. As λ(s,d) ≃ 4·10−4, whereas λ(b,d) ≃ 1·10−2 and
t t
λ(b,s) ≃ 4·10−2, we would roughly expect the deviation from the SM functions in the K system
t
to be by an order of magnitude larger than in the B system, and even by a larger factor than
d
in the B system.
s
This strong hierarchy in factors 1/λq1,q2 is only partially compensated by the opposite
t
hierarchy in the F functions [2], so that we expect that contributions of new physics are bigger
in the K decays than in the B decays, even if not by an order of magnitude. This expectation
will be confirmed by our numerical analysis in the following section.
5. Numerical Analysis
5.1. Breaking of Universality and New Sources of CP Violation
In the model discussed here the universality in the rare decays is generally broken, as clearly
seen in formulae (14) and (15).
Moreover, defining the total X functions as
i
XK = XSM +XsVd−A+XsVd = |XK|eiθXK , (16)
Xs = XSM +XbVs−A+XbVs = |Xs|eiθXs , (17)
Xd = XSM +XbVd−A+XbVd = |Xd|eiθXd , (18)
(19)
we find that these functions become complex quantities and their phases turn out to exhibit a
non-universal behaviour.
Numerically we find roughly
|X | |X | |X |
K s d
0.60 ≤ ≤ 1.30, 0.95 ≤ ≤ 1.08, 0.90 ≤ ≤ 1.12, (20)
X X X
SM SM SM
implying that the CP-conserving effects in the K system can be much larger than in the B and
d
B systems (as expected already in the previous section), where new physics effects are found
s
to be small.
We illustrate the first two quantities in (20) in Fig. 3, where we show the ranges allowed in
the space (|X |,|X |). The solid thick line represents the relation |X |= |X | that holds in the
K s s K
SM and more generally in models with Constrained Minimal Flavour Violation (CMFV) where
all flavour violation is governed by the CKM matrix and only SM operators are relevant [13]).
The crossing point of the thin solid lines indicate the SM value. The departure from the solid
thick line gives the size of non-CMFV contributions that are caused dominantly by NP effects
in the K system.
Figure 3. Breakdown of the universality
between |X | and |X |. Here and in
K s
the following we show in blue the points
which satisfy all the constraints on the Figure 4. Breakdown of the universality
∆F = 2 observables [3] and in orange the between θK and θs .
X X
points with in addition the requirement of
moderate fine-tuning in ε
K
(∆ (ǫ ) ≤ 20 [15]).
BG K
If we now investigate the new phases, i.e. new sources of CP violation, we find the ranges
−45◦ ≤ θK ≤ 25◦, −9◦ ≤ θd ≤ 8◦, −2◦ ≤ θs ≤ 7◦, (21)
X X X
implying that the new CP–violating effects in the ∆F = 1 b → dν¯ν and b → sν¯ν transitions are
very small, while those in K decays can be sizable (see also Fig. 4).
Alsofromthese lastresults itis evidentthat flavour universality can besignificantly violated.
5.2. Rare Decays of K and B Mesons
In this subsection we investigate the possible correlations between different ∆F = 1 and also
between∆F = 1and∆F = 2observables. IntheSMandinmodelswithCMFVtheraredecays
analysed in the present work depend basically on universal functions. Consequently, a number
of correlations exists between various observables not only within the K and B systems but also
between K and B systems. In this subsection we analyse how some of these correlations are
violated in the WED framework.
First, we study the correlation between the branching ratios of the theoretical very clean
decays K → π0ν¯ν andK+ → π+ν¯ν. InFig. 5weshow ourresult. We observethatbothdecays
L
candiffersimultaneouslyfromtheSMandinparticularthatthebranchingratioBr(K → π0ν¯ν)
L
can be as large as 15·10−11, that is by a factor of 5 larger than its SM value [14,17]
Figure 5. Br(K → π0ν¯ν) as a function
L
of Br(K+ → π+ν¯ν). The shaded area
represents the experimental 1σ-range for Figure 6. Correlation between Br(B →
Br(K+ → π+ν¯ν). The Grossman-Nir Xsνν¯) and Br(B → Xdνν¯). The black
bound [16] is displayed by the dotted line, line represents the universal CMFV result
while the solid line separates the two givenbythe ratio |Vtd|2 and the black point
areas where Br(K → π0ν¯ν) is larger or |Vts|2
L the SM prediction.
smaller than Br(K+ → π+ν¯ν). The dark
point shows the SM prediction.
Br(K → π0ν¯ν)SM = (2.8±0.6)·10−11, (22)
L
while being still consistent with the measured value for Br(K+ → π+ν¯ν). The latter branching
ratio can be enhanced by at most a factor of 2 but this is sufficient to reach the central
experimental value [18]
Br(K+ → π+ν¯ν)exp = (17.3+11.5)·10−11, (23)
−10.5
to be compared with the SM value [19]
Br(K+ → π+ν¯ν)SM = (8.5±0.7)·10−11. (24)
Unfortunately, if we do the same type of analysis for the inclusive decays of B mesons
B → X ν¯ν and B → X ν¯ν, we do not find the same spectacular deviation from the SM
d s
and in general from models with CMFV.
Of interest is the ratio
Br(B → X νν¯) |V |2
d td
= ·P , (25)
Br(B → Xsνν¯) |Vts|2
where
|XV−A+XV/2|2 +|XV/2|2
P ≡ d d d . (26)
|XV−A+XV/2|2 +|XV/2|2
s s s
In the SM and models with CMFV we have a clear correlation between these branching
ratios: P = 1. Instead in WED we have deviations from this correlation, as seen in Fig. 6. In
particular we find the range
0.93 ≤ P ≤ 1.07. (27)
This result shows that NP effects in rare B decays are significantly smaller than in rare K
decays as already expected fromouranatomy of NP effects in Sec.4.1. InB decays theresulting
deviation is small and will be difficult to measure.
As conclusion of this section we investigate one of the possible correlations between K and B
decays (the two branching ratios of K+ → π+ν¯ν and B → µ+µ−) and one between observables
s
with ∆F = 1 and ∆F = 2 (the branching ratio of K+ → π+ν¯ν and the asymmetry S ).
ψφ
Figure 7. Br(Bs → µ+µ−)/Br(Bs → Figure 8. Br(K+ → π+νν¯) as
µ+µ−)SM as a function of Br(K+ → a function of S . The shaded area
ψφ
π+ν¯ν). The shaded area represents the represents the experimental 1σ-range for
experimental 1σ-range for Br(K+ → Br(K+ → π+νν¯). The dark point shows
π+ν¯ν) and the dark point shows the SM the SM prediction.
prediction.
In Fig. 7 we show that the branching ratio Br(B → µ+µ−) can deviate only by 15% from
s
theSMvalue, whilemorepronouncedeffects (200%) arepossibleinBr(K+ → π+ν¯ν)aswehave
seen before. When Br(K+ → π+ν¯ν) is sizably enhanced, it is almost impossible to distinguish
the branching ratio Br(B → µ+µ−) from the SM prediction.
s
Instead in Fig. 8, we observe that it is possible to obtain a large contribution of new physics
for both the observables S and the branching ratio Br(K+ → π+νν¯), but not simultaneously.
ψφ
In conclusion, Fig. 7 and Fig. 8 show that, despite of the many free parameters of the model,
oncethatwehavefixedallthe∆F = 2observablesandinparticularǫ ,themodelpredictssome
K
precise patterns of flavour violation that can be confirmed or ruled out by future experiments.
In particular, if future experiments will show a simultaneous enhancement for both S and
ψφ
Br(K+ → π+νν¯) or big enhancements for rare B decays, the model will be ruled out.
6. Conclusions
In the first part of this writing we have presented some of the reasons for which warped extra
dimensional models could be a competitor of supersymmetry. In particular we showed that,
in addition to addressing the gauge hierarchy problem, they can also naturally explain the
hierarchies between masses and mixings of the quarks and leptons of the SM.
In the second part we have analysed the most interesting rare decays of K and B mesons in
a warped extra dimensional model with a custodial protection of flavour diagonal and flavour
non-diagonal Z boson couplings to left-handed down quarks.
Once that we have fixed the scale of new physics to a scale that is in the reach of LHC
((2−3)TeV),weconsideronlythepointsoftheparameterspacewhichrespectalltheconstraints
from ∆F = 2 observables, and in particular from ǫ . Studying this region of parameter space
K
we arrive to these main conclusions:
In the future experiments, an observation of a large S would, in the context of the model
ψφ
considered here, preclude sizable NP effects in rare K decays. On the other hand, finding
S to be SM-like will open the road to large NP effects in rare K decays. Independently of
ψφ
the experimental value of S , NP effects in rare B decays are predicted to be small and an
ψφ
observationof large departures fromSMpredictions infuturedata wouldputthe model considered
here in serious difficulties.
Acknowledgments
Ithanktheotherauthorsof[2]: M.Blanke, A.J.Buras,B.DulingandK.GemmleraswellasW.
Altmannshofer and A.J. Buras for the careful reading of this writeup. This work was partially
supportedbytheEuropeanCommunitysMarieCurieResearchTrainingNetwork undercontract
MRTN-CT-2006-035505 (HEPTOOLS).
References
[1] RandallL and SundrumR 1999 Phys. Rev. Lett. 83 3370–3373 (Preprint hep-ph/9905221)
[2] BlankeM, Buras A J, DulingB, Gemmler K and Gori S 2008 (Preprint 0812.3803)
[3] BlankeM, Buras A J, DulingB, Gori S and Weiler A 2008 (Preprint 0809.1073)
[4] Gherghetta T and Pomarol A 2000 Nucl. Phys. B586 141–162 (Preprint hep-ph/0003129)
[5] DulingB 2009 (Preprint 0901.4599)
[6] Agashe K,Delgado A,May M J and SundrumR 2003 JHEP 08 050 (Preprint hep-ph/0308036)
[7] Csaki C, Grojean C, Pilo L and Terning J 2004 Phys. Rev. Lett. 92 101802 (Preprint hep-ph/0308038)
[8] Agashe K,Contino R,Da Rold L and Pomarol A 2006 Phys. Lett. B641 62–66 (Preprint hep-ph/0605341)
[9] Casagrande S, Goertz F, Haisch U, NeubertM and Pfoh T 2008 JHEP 10 094 (Preprint 0807.4937)
[10] AlbrechtM, Blanke M, Buras A J, DulingB and Gemmler K in preparation
[11] Hall L J and Randall L 1990 Phys. Rev. Lett. 65 2939–2942
[12] D’Ambrosio G, Giudice G F, Isidori G and Strumia A 2002 Nucl. Phys. B645 155–187 (Preprint
hep-ph/0207036)
[13] Buras A J, Gambino P, Gorbahn M, Jager S and Silvestrini L 2001 Phys. Lett. B500 161–167 (Preprint
hep-ph/0007085)
[14] Buras A J, Schwab Fand Uhlig S 2008 Rev. Mod. Phys. 80 965–1007 (Preprint hep-ph/0405132)
[15] Barbieri R and Giudice G F 1988 Nucl. Phys. B306 63
[16] Grossman Y and Nir Y 1997 Phys. Lett. B398 163–168 (Preprint hep-ph/9701313)
[17] Mescia F and Smith C 2007 Phys. Rev. D76 034017 (Preprint 0705.2025)
[18] Artamonov A V et al. (E949) 2008 Phys. Rev. Lett. 101 191802 (Preprint 0808.2459)
[19] Brod J and Gorbahn M 2008 Phys. Rev. D78034006 (Preprint 0805.4119)